electronic reprintJournal of

SynchrotronRadiation

ISSN 0909-0495

Editors: G. Ice, A. Kvick and T. Ohta

Coherent X-ray diffraction investigation oftwinned microcrystals

Miguel A. G. Aranda, Felisa Berenguer, Richard J. Bean, Xiaowen Shi,Gang Xiong, Stephen P. Collins, Colin Nave and Ian K. Robinson

J. Synchrotron Rad. (2010). 17, 751–760

Copyright c© International Union of Crystallography

Author(s) of this paper may load this reprint on their own web site or institutional repository provided thatthis cover page is retained. Republication of this article or its storage in electronic databases other than asspecified above is not permitted without prior permission in writing from the IUCr.

For further information see http://journals.iucr.org/services/authorrights.html

Synchrotron radiation research is rapidly expanding with many new sources of radiationbeing created globally. Synchrotron radiation plays a leading role in pure science andin emerging technologies. The Journal of Synchrotron Radiation provides comprehensivecoverage of the entire field of synchrotron radiation research including instrumentation,theory, computing and scientific applications in areas such as biology, nanoscience andmaterials science. Rapid publication ensures an up-to-date information resource for sci-entists and engineers in the field.

Crystallography Journals Online is available from journals.iucr.org

J. Synchrotron Rad. (2010). 17, 751–760 Miguel A. G. Aranda et al. · Twinned microcrystals

research papers

J. Synchrotron Rad. (2010). 17, 751–760 doi:10.1107/S0909049510039774 751

Journal of

SynchrotronRadiation

ISSN 0909-0495

Received 23 August 2010

Accepted 5 October 2010

# 2010 International Union of Crystallography

Printed in Singapore – all rights reserved

Coherent X-ray diffraction investigation oftwinned microcrystals

Miguel A. G. Aranda,a* Felisa Berenguer,b Richard J. Bean,b Xiaowen Shi,b

Gang Xiong,b Stephen P. Collins,c Colin Navec and Ian K. Robinsonb,c*

aDepartamento de Quımica Inorganica, Cristalografıa y Mineralogıa, Universidad de Malaga,

29071 Malaga, Spain, bLondon Centre for Nanotechnology, University College, 17–19 Gordon

Street, London WC1H 0AH, UK, and cDiamond Light Source Ltd, Diamond House, Harwell

Science and Innovation Campus, Didcot, Oxfordshire OX11 0DE, UK. E-mail: g_aranda@uma.es,

i.robinson@ucl.ac.uk

Coherent X-ray diffraction has been used to study pseudo-merohedrally

twinned manganite microcrystals. The analyzed compositions were Pr5/8Ca3/8-

MnO3 and La0.275Pr0.35Ca3/8MnO3. The prepared loose powder was thermally

attached to glass (and quartz) capillary walls by gentle heating to ensure

positional stability during data collection. Many diffraction data sets were

recorded and some of them were split as expected from the main observed twin

law: 180� rotation around [101]. The peak splitting was measured with very high

precision owing to the high-resolution nature of the diffraction data, with a

resolution (�d/d) better than 2.0 � 10�4. Furthermore, when these microcrystals

are illuminated coherently, the different crystallographic phases of the structure

factors induce interference in the form of a speckle pattern. The three-

dimensional speckled Bragg peak intensity distribution has been measured

providing information about the twin domains within the microcrystals.

Research is ongoing to invert the measured patterns. Successful phase retrieval

will allow mapping out the twin domains and twin boundaries which play a key

role in the physical properties.

Keywords: coherent diffraction; CXDI; pseudo-merohedral twinning; perovskites;manganites.

1. Introduction

Twinning is a crystal growth anomaly in which the specimen is

composed of several distinct domains whose orientations are

related by one or more symmetry elements that are not a part

of the space-group symmetry of the single crystal, i.e. related

by the so-called twin operator(s) or twin law(s) (Cahn, 1954;

Santoro, 1974; Yeates, 1997). Multiple-crystal growth dis-

orders are common, but twinning refers to special cases where

the extra symmetry element(s) must be of the kind encoun-

tered in crystal morphology (a centre of symmetry, a mirror

plane or a rotation axis). If the extra symmetry element is a

mirror plane, called a twin plane, then this plane must be

parallel to a lattice plane of the same d-spacing in both

domains. If the extra symmetry is a rotation axis, called a twin

axis, then this axis must be parallel to a lattice row common to

both domains (Cahn, 1954; Koch, 1992). The twinning occur-

rence is fairly common in crystals of inorganic materials,

organic compounds and macromolecular specimens, and it can

be a serious complication in the crystal structure determina-

tion process.

Several different categories of twinning can be defined

according to the coincidence of the separated lattices

(Buerger, 1960; Yeates, 1997). The situation in which the

lattice overlap occurs in two (or one) dimensions is known as

non-merohedral or epitaxial twinning. The case in which the

lattices of the domains coincide exactly in three dimensions is

known as merohedral twinning. Hemihedral twinning is a

special subclass where there are just two domain orientations.

The cases where the lattices overlap approximately, but not

exactly, in three dimensions are referred to as pseudo-mero-

hedral twinning. This requires unusual unit-cell geometries

that yield a lattice with pseudo-symmetry which is higher than

the point-group symmetry (for instance the case of ortho-

rhombic perovskites with a ’ c, both edges deriving from the

cubic parent structure).

Perovskite materials display many interesting properties

including ferromagnetism, piezo- and ferro-electricity, multi-

ferroic behaviour, etc. The archetypal perovskite structure is

cubic but, in order to fine-tune the physical properties, struc-

tural distortions may take place with many materials being

orthorhombic or monoclinic. Low-symmetry perovskite crys-

electronic reprint

tals therefore fulfil the requisites for pseudo-merohedry and

they are commonly twinned. Twinning in orthorhombic

perovskites has been extensively studied by a number of

techniques including transmission electron microscopy (TEM)

(White et al., 1985; Keller & Buseck, 1994; Hervieu et al.,

1996), precession electron diffraction (Ji et al., 2009), photo-

emission spectroscopy (Sarma et al., 2004), neutron single-

crystal diffraction (Daoud-Aladine et al., 2002) and X-ray

single-crystal diffraction (Van Aken et al., 2002). Furthermore,

Pr0.5Ca0.5MnO3 twinned crystals were studied by synchrotron

(micro)diffraction to map the domains (Turner et al., 2008) but

the coherence properties of the beam were not fully exploited

to provide information about the crystal structure in real

space. To the best of our knowledge there are no other

reported works dealing with the study of twinned crystals by

coherent X-ray diffraction.

Here we apply the method of coherent X-ray diffraction

imaging (CXDI), which is a rapidly advancing form of lensless

microscopy (Neutze et al., 2000; Vartanyants et al., 2007; Jiang

et al., 2008; Thibault et al., 2008; Nishino et al., 2009; Robinson

et al., 2010, and references therein) that was opened up by the

realisation that oversampled diffraction patterns may be

inverted to obtain real-space images. The possibility was first

pointed out by Sayre (1952) and demonstrated by Miao et al.

(1999). The phase information of the diffraction pattern,

which is lost in the intensity measurement, is embedded in a

sufficiently sampled coherent diffraction pattern, which may

allow the inversion of the diffraction data set back to an image

by computational methods. In addition to obtaining three-

dimensional images of studied nanoparticles, CXDI may give

different types of information including the quantitative strain

field inside a crystal (Pfeifer et al., 2006; Robinson & Harder,

2009; Newton et al., 2010).

In this paper we focus on the use of CXDI for studying

twinned microcrystals. When such a crystal is illuminated

coherently, speckle patterns can form owing to the differences

in amplitude and/or phase of the structure factors for different

twin domains, and also from the continuous phase changes

between them. Here, we show that this three-dimensional

speckled Bragg peak distribution can be measured providing

information about the twin domains within the crystals.

Research is ongoing to invert the patterns to obtain three-

dimensional images of the twin distribution. Three-dimen-

sional images of the twinned domains can provide information

relevant to the properties of materials. An ultimate goal is

that, by separating out each component, more accurate

diffraction intensities may be obtained which would help in

the structure determination process of twinned crystals.

2. Experimental section

Three manganite compositions were synthesized by the

ceramic method as previously reported (Collado et al., 2003).

The stoichiometries were Pr5/8Ca3/8MnO3 , La5/8Ca3/8MnO3

and La0.275Pr0.35Ca3/8MnO3 , and they are hereafter abbre-

viated PCMO, LCMO and LPCMO, respectively. The three

samples were characterized by high-resolution laboratory

X-ray powder diffraction at room temperature. Powder

patterns were collected on a Philips X’Pert Pro MPD

diffractometer equipped with a Ge(111) primary mono-

chromator (strictly monochromatic Cu K�1 radiation) and an

X’Celerator detector. Transmission electron microscopy

characterization was carried out using a Philips CM-200

microscope with the powders dispersed on copper grids.

The sample preparation step for a CXDI experiment is

important as the positional stability of the particles within the

beam must be ensured. In order to carry out this type of

experiment for loose powders, we have developed a simple

method which is valid for thermally stable samples. Dispersed

crystals were attached to borosilicate glass (and also quartz)

capillaries by gentle heating in a butane flame (using a cigar-

ette lighter) up to the point where the capillary glass softens.

The results are better when the heating is applied to vertically

arranged capillaries (as horizontally aligned capillaries are

prone to severe bending). Fig. 1 shows an optical micro-

photograph of a borosilicate glass capillary with the PCMO

loose powder thermally attached to the wall.

Coherent X-ray diffraction patterns for selected manganite

microcrystals were collected at beamline I16, Diamond Light

Source, which is equipped with a six-circle kappa goniometer

at 50 m from the source. The wavelength, � = 1.55 A (E =

8.00 keV), was selected using a Si(111) channel-cut mono-

chromator. The beam was horizontally focused down to

�4 mm by using a Kirkpatrick–Baez (KB) mirror pair. The

vertical and horizontal dimensions of the beam before these

mirrors, �30 mm, were selected by the final beamline slits

before the KB mirrors; these slits are not very precise,

however. To ensure this beam was reasonably coherent, a

300 mm horizontal slit was placed in front of the beamline

focusing mirror (which remained in place in the beamline).

This slit is located 27 m from the source and the diffractometer

is at 50 m. The horizontal coherence length at the sample is

therefore increased to �D/d = 12 mm, assuming stochastic

illumination of the intermediate 27 m slit. The sample was

aligned by scanning the X-ray absorption of the capillary with

a point detector. After the alignment of the capillary, the

appropriate Bragg peaks, for instance (121) at 33.17� 2� for

PCMO, were identified by using a Pilatus 100K detector

research papers

752 Miguel A. G. Aranda et al. � Twinned microcrystals J. Synchrotron Rad. (2010). 17, 751–760

Figure 1Optical microphotography of dispersed PCMO microcrystals attached toa glass capillary (diameter = 0.3 mm).

electronic reprint

(172 mm � 172 mm pixel-size photon counting detector)

permanently mounted in the detector arm. After centring the

reflection, this detector was moved away and the reflection

was again centred in a Princeton Instruments CCD detector

with optical coupling to a columnar CsI scintillator. The pixel

size was 20 mm � 20 mm and it was placed far from the sample,

at 1220 mm, to provide data with sufficient angular resolution.

Two-dimensional slices were recorded as the Bragg peak was

rocked through the Ewald sphere by rotating the sample

within the beam, usually in 0.005� step size (� rocking angle)

for about 0.6� although this number depends upon the width

of the analyzed peak which was previously pre-scanned at

lower resolution. For every discussed scan, the exposure time,

number of accumulations and maximum intensity is given

below. Coherent diffraction patterns were obtained by

collating the slices of the high-resolution scan to form

complete three-dimensional diffraction data sets. The pixel

binning size and rocking step size were chosen to ensure at

least three pixels per fringe, to ensure the data were over-

sampled.

The coherence of a beam of light is generally described by

two components, the transverse and longitudinal coherence

lengths, �T and �L, respectively (Born & Wolf, 1999; Leake et

al., 2009). The transverse coherence is dependent on the size

(S) of the source/undulator itself, and it is given by �T = �D/S,

where D is the distance from the source to the sample. For a

typical third-generation synchrotron beamline, the raw trans-

verse coherence is �10 mm and �100 mm for the horizontal

and vertical directions, respectively. By closing the beamline

mid-point slit (see above), the horizontal transverse coherence

length was raised to �12 mm, sufficient to give some limited

degree of coherence beyond the �30 mm sample slits. On the

other hand, the longitudinal coherence is dependent on the

bandwidth of the monochromator, �L = �2/2��. For a silicon

(111) monochromator, �L ’ 0.6 mm. This couples to the optical

path length difference of rays through the sample. Therefore,

when the crystal size is smaller than the coherence lengths, the

sample is said to be in the coherent limit and it meets the

required conditions for CXDI measurements. The fringe

visibility can be lost by violation of any of the three coherence

requirements.

3. Results

Laboratory powder diffraction data for the three manganite

samples showed that they were microcrystalline single phase

powders. The three phases were indexed in an orthorhombic

distorted unit cell of Pnma symmetry, with edges ’ ffiffiffi2

pac � 2ac

� ffiffiffi2

pac where ac stands for the basic perovskites cubic unit-

cell parameter, in agreement with previous results (Uehara et

al., 1999; Collado et al., 2003). The unit-cell values for the three

phases are given in Table 1. It is clear from the unit-cell edges

given in Table 1 that these manganites are very prone to

pseudo-merohedral twinning.

The sizes, shapes and variability of the manganite particles

were studied by TEM. Fig. 2 displays bright and dark field

microphotographs for selected particles of PCMO. It must be

noted that the dispersion in particle sizes is large, from small

particles of about 0.2 mm to large particles longer than 1 mm.

The shapes were also variable but most of the observed

particles were elongated prisms (see Fig. 2).

Fig. 3 shows the diffraction signal of PCMO and LPCMO

microcrystals, collected in the Pilatus detector, as an example.

Only those well shaped rounded reflections isolated from any

other diffracting signal were centred, for instance the (210)

reflection of LPCMO in Fig. 3(b).

Three main types of coherent diffraction data sets were

recorded on the CCD: (i) single centred, (ii) doubled-centred

and (iii) multi-centred peaks. Fig. 4 (top) shows an example of

each type of data set as two-dimensional slides. The angles

used for the discussion of the results (see below) are also

shown in Fig. 4(b). Fig. 4 (bottom) displays views of the three-

dimensional integrated patterns for the three types of micro-

crystals in order to highlight the widths of the measured

diffraction peaks. Eleven good quality data sets were collected

for PCMO and four data sets for LPCMO. It must be high-

research papers

J. Synchrotron Rad. (2010). 17, 751–760 Miguel A. G. Aranda et al. � Twinned microcrystals 753

Table 1Unit-cell parameters for Pr5/8Ca3/8MnO3, La0.275Pr0.35Ca3/8MnO3 andLa5/8Ca3/8MnO3 determined by X-ray powder diffraction with alaboratory source.

PCMO LPCMO LCMO

a (A) (ac) 5.443 (3.849) 5.441 (3.848) 5.448 (3.853)b (A) (ac) 7.676 (3.838) 7.676 (3.838) 7.694 (3.847)c (A) (ac) 5.421 (3.834) 5.440 (3.847) 5.462 (3.863)Sac (%) +0.41 0.02 �0.26��ac (�) 0.232 0.011 0.147�2�(200) (�) 0.140 0.010 0.091

Figure 2TEM bright (left) and dark (right) field micrographs showing twoselected particles of PCMO.

electronic reprint

lighted that we could not collect CXD data for any of the

LCMO microcrystals we tried. The diffraction signal dis-

appears from the CCD detector when centring the crystal. We

speculate that the photoelectric effect may be playing an

important role as this is the highest electrically conducting

sample. Therefore, LCMO will not be further discussed in

this work.

4. Real-space phase model of twinning

PCMO has orthorhombic space group Pnma, with lattice

constants given in Table 1, which is a supercell of the simple

cubic perovskite. These perovskites are known to show three

main laws for pseudo-merohedral twinning: a 180� rotation

around [101], 90� around [101] and a 180� rotation around

[121] (Wang & Liebermann, 1993; Keller & Buseck, 1994).

The ‘a–c’ twinning arises from the accommodation of the small

difference between the a and c lattice constants, possibly

forming upon cooling the crystals from a higher symmetry

phase at elevated temperature. Fig. 5 shows an exaggerated

view of the matching of the {101} planes at the twin boundary,

running up the figure. It is clear that the a–c lattice difference

corresponds to a small rotation of the twin domains, giving two

sets of Bragg peaks for all reflections except along the hkh

symmetry axis.

research papers

754 Miguel A. G. Aranda et al. � Twinned microcrystals J. Synchrotron Rad. (2010). 17, 751–760

Figure 4(Top) Coherent diffraction patterns of PCMO crystals showing a single peak with fringes (PCMO1–9, frame 77; 30 s of exposure time, one accumulation,Imax = 34600 counts s�1) (a), a split Bragg peak with speckles extending between the two centres (PCMO3–7, frame 28; 5 s of exposure time, threeaccumulations, Imax = 71630 counts s�1) (b) and a multi-centred data set (PCMO2–13, frame 88; 30 s of exposure time, one accumulation, Imax =4260 counts s�1) (c). The full CCD recording area is shown for the sake of comparison. The angles discussed in the paper (�, � and 2�) are also displayed.(Bottom) Projections of the three-dimensional ‘collated’ diffraction patterns showing the overall widths of the diffraction peaks.

Figure 3Microcrystal two-dimensional diffraction images of PCMO (a) and LPCMO (b) collected on the Pilatus detector. The Rietveld-fitted powder diffractionpatterns of the corresponding regions are also shown to highlight the width of the regions where diffraction is expected.

electronic reprint

If the domains are small, their Bragg peaks will be enlarged

owing to the classical ‘finite size’ effect. If the domains are

sufficiently small, the two members of the split Bragg peaks

will overlap and, assuming the beam is coherent over the size

of the domains, will lead to interference effects in the overlap

region of the diffraction pattern. Each of the two peaks will be

speckled owing to the domain arrangement within the illu-

minated piece of crystal and the speckles will combine toge-

ther in the interference pattern. It is a general rule, for domain

structures at least, that the number of speckles in the peak is

approximately equal to the number of domains in the beam.

The rule assumes the domains are independent, with large

phase shifts between them, and applies in one, two or three

dimensions. The rule can be derived by noting that the size of a

speckle is the reciprocal of the size of the illuminated area,

while the width of the speckle distribution is the reciprocal of

the size of a typical domain.

The words ‘speckle’ and ‘fringe’ are used rather inter-

changeably in the literature. Fringes appear in optical inter-

ferometers as regular linear arrays of constructive and

destructive interference, and are used as a way to measure

coherence, among other things. Fringes also arise in the shape-

diffraction patterns of simple objects, parallel from pairs of

edges or circular from round apertures or scatterers. Speckles,

on the other hand, are the result of random interference

between a large number of waves with arbitrary, yet well

defined, amplitudes and phases. In principle their degree of

modulation can be used to measure coherence, as reported by

Alaimo et al. (2009) using the dynamical near-field speckles

formed by scattering from colloidal particles. The inherently

random nature of domain structures owing to twinning are

therefore expected to give rise to speckles rather than simple

fringes. Here, we use the term ‘speckle’ to describe the fluc-

tuation in the Bragg peak intensity distribution owing to

interference between the (smaller) twin domains, and the

term ‘fringe’ to mean the structure owing to the (larger)

crystallites. If the domain structure exists across the entire

crystal, the spacing between the speckles will be determined

by the crystal size. However, the overall peaks will have a

breadth dependent on the domain size with the number of

speckles in the peak related to the number of domains in the

crystal. The formation of both speckles and fringes is a

necessary consequence of having some degree of coherence in

the beam.

There is considerable merit, where possible, in modelling

the diffracting object as the product of a periodic structure

(the ideal crystal) and a macroscopic modulation of the phase

that characterizes the structures of interest (i.e. twin domains).

In reciprocal space, such a multiplication becomes a convo-

lution, and the motif of the macroscopic object is repeated

around each point in reciprocal space. In the present case,

such a treatment is complicated by the presence of two lattices.

We would like to interpret the combined double Bragg peak,

along with these speckles, in terms of the Fourier transform of

a single object in real space. We therefore make use of the

shifting property of the Fourier transform: when the object in

real space is multiplied by a linear phase ramp, the diffraction

pattern is shifted by an amount proportional to the ramp. In

this way the two Bragg peaks of the split pair, shifted in

opposite directions, can be attributed to separate regions of

the crystal overlaid with opposite-directed phase ramps. This

is illustrated in Fig. 5(a), where the independent lattices of the

two separate domains can be described as a transformation

from a single reference lattice. If the origin is chosen at the

corner of the unit cells meeting at the twin plane, each unit cell

is shifted by an amount proportional to its distance from the

twin plane, or a linear function of its distance. This shift gives

rise to a phase in the diffraction that increases linearly with

distance from the twin plane, in other words a phase ramp.

Therefore, through the Fourier shift theorem, the two domains

give rise to opposite peak shifts in the combined diffraction

pattern, seen as the splitting.

This picture is easily generalized to the full mosaic domain

structure of an arbitrarily twinned crystal, which is thereby

mapped onto a complex image of unit amplitude and spatially

varying phase. All the left-rotated domains have phase ramps

with negative slope and all the right-rotated domains have

phase ramps with positive slope; where they join, the phase

should be continuous (although their derivatives are not),

assuming they meet at a complete unit-cell boundary, but the

slope reverses. The picture can be readily generalized to three

dimensions: the macroscopic real-space phase is simply the

three-dimensional displacement of the atoms from those of

the ideal reference lattice, projected onto the Q-vector of

the reflection under consideration. The choice of ‘reference’

research papers

J. Synchrotron Rad. (2010). 17, 751–760 Miguel A. G. Aranda et al. � Twinned microcrystals 755

Figure 5(a) Exaggerated schematic view of a pseudo-merohedral twin pair owingto 180� rotation around [101] for orthorhombically distorted perovskite.(b) Calculated diffraction spots illustrating the splitting of the Braggpeaks. (c) Two-dimensional model of the two domains, each containing9 � 9 unit cells, with the difference between the a and c dimensions beingten times greater than is the case for the PCMO crystals in order tomagnify the effects for demonstration purposes. (d) Calculated scatteringpattern using the nearBragg program from the two-dimensional twinnedcrystal given in (c).

electronic reprint

lattice is arbitrary; it simply defines the reciprocal lattice with

respect to which the peak shifts are observed.

The program nearBragg (http://bl831.als.lbl.gov/~jamesh/

nearBragg/) from James Holton was used for some initial

simulations. The following description is extracted from the

nearBragg manual: “ . . . nearBragg calculates the distance

from one or more source points to each ‘atom’ and then from

the ‘atom’ to the centre of a detector pixel. The sin and cos of

2� times the number of wavelengths involved in this total

distance is then added up and the amplitude and phase of the

resultant wave from the whole sample (and the whole source)

are obtained for each pixel. The intensity at each pixel is the

modulus-square of the amplitude. The atoms are considered

point scatterers with an intrinsic structure factor of ‘1’ in all

directions . . . ”. Results from one of the simulations are shown

in Fig. 5(d). This type of simulation is easy to relate to the

schematic in Fig. 5(a) and it is shown for this reason. However,

the phase ramp description gives the same results [as will be

shown below in Fig. 6(a)] and it is computationally much faster

for three-dimensional objects. The phase ramp description is

therefore used for the remaining simulations.

To illustrate the effect of the opposite phase ramps on the

diffraction patterns, in Fig. 6 we present two simulated three-

dimensional objects with a double phase ramp. The calcula-

tions are carried out using a fast Fourier transform on a 256 �256 � 128 array and an object of 48 � 24 � 24 grid points, such

that each simulated twin domain has a cubic shape. The

amplitudes are 1 within the object and 0 outside. The intro-

duced phase ramps within the object are shown on a colour

wheel with green representing ’ = 0, blue ’ = �� and red ’ =

+�. The corresponding amplitudes of the three-dimensional

diffraction pattern are depicted at the bottom of the figure

showing the splitting of the peak owing to the phase structure

and the intermodulation of the diffraction in between. The

magnitude of the ramps, 1.5� and 2�, in Figs. 6(a) and 6(b),

respectively, corresponds to an increasing angular separation

of the peaks (see Fig. 6, bottom). Since the fringe spacing is

fixed by the crystal size, the result is an increasing number of

fringes connecting the two centres of the Bragg peaks as the

slope of the ramps increases (Fig. 6b).

So far we have omitted the structure factor from this

discussion, but it is straightforward to introduce that too. In

the perovskite case we have been considering that the two

member peaks of the pair will have roughly the same structure

factor, even if they have different indices, because they are

attributed to the same reflection of the cubic parent structure.

In the general case where different unrelated structure factors

mix together, this must be built into the complex object whose

Fourier transform gives the full diffraction (speckle) pattern.

Both the amplitude and phase of the structure factor are

attributed to the regions of space occupied by the domain,

multiplied by the phase ramp function corresponding to the

domain rotation relative to the reference lattice. This picture

applies equally to the interpretation of data: following phase

retrieval of the diffraction pattern, the inverted three-dimen-

sional image of the crystal can be interpreted directly in terms

of local structure factors and domain rotations (ramps).

It is known that phase retrieval is difficult in this general

case and methods still need to be developed. The best methods

to date, based on Fienup’s (1982) ‘hybrid input–output’

method, still require a ‘real-space constraint’. The description

above could conceivably be coded into a suitable algorithm,

for example exploiting the slow-varying continuous property

of the real-space phase image. Such a method has been used

successfully by Minkevich et al. (2007) for analysing strains in

semiconductor heterostructures (without twinning).

However, if phase retrieval and imaging are not possible,

limited information can be extracted from the diffraction

pattern itself. The speckle contrast, usually defined as the

‘visibility’ (Imax � Imin)/(Imax + Imin), is a rough estimate of the

range of real-space phases present in a diffracting object. If the

object is a pure, but weak, phase object, illuminated by a soft-

edged beam, then it can be shown, by a vector summation

diagram representing the Fourier transform, that the speckle

contrast is proportional to the range of phases present in the

object, envisaged as a real-space phase map in three dimen-

sions. The speckle contrast can be estimated experimentally as

the variance of the intensity over parts of the diffraction

pattern. There is a potentially important application to the

case of merohedral twinning discussed above. When domains

of different structure factor but identical lattice are present in

a twinned crystal, the speckle visibility will be proportional to

the crystallographic (complex) amplitude difference between

the two structure factors. This might have some potential

in crystallographic phase measurement. In other cases, the

research papers

756 Miguel A. G. Aranda et al. � Twinned microcrystals J. Synchrotron Rad. (2010). 17, 751–760

Figure 6(Top) Two bicrystals shown as a translucent three-dimensional box withcolours representing the phase change. The two twins (A and B) have thesame dimension (24 � 24 � 24 grid points) but different slopes of thephase ramps: (a) 1.5�, (b) 2�. (Middle) Calculated three-dimensionalcoherent diffraction patterns of the two twinned objects. (Bottom)Isoscalar plane cut of the diffraction patterns.

electronic reprint

speckle contrast will be due to internal grain boundaries

(density discontinuities) within the sample. For pseudo-

merohedral twinning, each component of the split Bragg peak

receives its main contribution from domains with only one of

the two orientations. These domains might be separated from

each other by ‘unseen’ domains with the alternative orienta-

tion, an extreme example of a density discontinuity. This is the

probable reason why a relatively high speckle contrast is

observed for the manganites studied here.

The important result of this section is that we can represent

the complicated array of twin domains within a crystal, in

which the lattice is separately defined for each domain, in

terms of a phase field over the entire object. The relative

positions of the domains are mapped onto phase shifts, while

the local rotations are mapped onto phase ramps. This method

is generally applicable, but might have little useful meaning in

the limit of very small domains or the case of partially crys-

talline or amorphous materials.

5. Discussion

Firstly, the data shown in Fig. 3, which are representative of

many recorded images, deserve a brief discussion. PCMO, as

shown in Table 1, has a much larger orthorhombic ac distor-

tion than LPCMO. This may be quantified by the lattice strain

in the ac plane, Sac (%),

Sac ¼2ða� cÞðaþ cÞ � 100; ð1Þ

and the corresponding values are given in Table 1. The larger

distortion for PCMO is also evident in Fig. 3 as the (200) and

(002) Bragg reflections are partially resolved in the powder

diffraction pattern. For LPCMO, those reflections are not

resolved in the powder pattern, owing to the very low

orthorhombic distortion, but some studied microcrystals

showed very large peak widths. The data shown in Fig. 3 for

LPCMO are indicative of a very large unit-cell parameter

distribution for some microcrystals. This may be due to

inhomogeneities in the cation distribution of this solid solution

but it may also be due to large microstrain owing to the rich

twin structure of these compositions. This observation may be

much related to its very complex low-temperature behaviour

displaying mesoscopic phase separation (Uehara et al., 1999)

and persistent magnetoresistive memory effect (Levy et al.,

2002).

Now we discuss the results obtained in the coherent high-

resolution diffraction patterns. We will focus on split Bragg

reflections and we will not consider in this work the unsplit

peaks as they are not likely to arise from twinned regions of

the manganite microparticles. Most of this study was dedicated

to recording data around the 2� ’ 33.1� region as it contains

the most intense diffraction peaks of these manganites: (200),

(121) and (020) (see Fig. 3). It must be noted that these

reflections cannot be distinguished by their measured 2�values as they have almost the same d-spacings, with their

angular positions slightly dependent upon the centring of the

microcrystal in the beam. However, their splitting behaviour

owing to the ac twinning is quite different (see Fig. 5).

Firstly, let us consider a (121) reflection. The 2� Bragg angle

for the pair arising from the two twin domains must have the

same value but they will be rotated by an angle, ��ac, defined

as

��ac ¼ ð�=2Þ � 2 tan�1ða=cÞ or ��ac ’ 1 � a=c: ð2ÞThe theoretical values for this splitting angle are also given in

Table 1. For PCMO, we have collected three data sets with

split peaks with an average ��ac value of 0.23 (1)�, which were

measured from the diffraction data as detailed in equation (3)

(see Fig. 4b),

�� ¼ ��2 þ��2� �1=2

: ð3ÞThis value is in excellent agreement with the expected angle

between the axes of the two twin individuals (see Table 1). It

must be noted that the measured 2� angles for two centres

were the same for this type of reflection. An example of this

type of double-centred coherent diffraction data set is shown

in Fig. 4(b). This misfit angle has been previously reported

(see, for instance, Wang & Liebermann, 1993) but the value

was larger (�1�) because the lattice strain in the ac plane for

the studied natural perovskites, CaTiO3, is larger.

Secondly, let us consider a (200) [or (002)] type reflection. It

is clear that, for a single domain crystal, if one reflection is

observed on the detector the other cannot be measured as

it is perpendicular. However, when the (200) reflection of a

pseudo-merohedrally twinned microcrystal is being measured,

the (002) diffraction from the other twin domain will also be

present close to that diffraction angle. The 2� difference

between the two centres of the peaks will be given by the

difference in the d-spacing values of those planes. So, for the

(200)/(002) peaks, we can define �2�(200) which will be the

difference of the diffraction angles of the (200) and (002) twin-

related diffraction peaks. The values of this angle, for the

studied manganite compositions, are also given in Table 1. The

angle has been measured for two PCMO microcrystals just by

converting the �y pixel distance to 2� angle, see Fig. 4(b)

(using the sample-to-detector distance). Fig. 7(a) shows one

example of this type of reflection, and the measured �2�(200)

angle, 0.15�, is in very good agreement with the expected one,

0.14� (see Table 1). Furthermore, these centres are also

displaced in � and � angles (see Fig. 4b). We can determine the

��(200) angle, 0.106�, which was also in very good agreement

with the expected one, (1 � a/c)/2 = 0.116�.

Furthermore, the (200)/(002) splitting described just above

can be generalized to any reflections having h 6¼ l indexes. We

were able to collect coherent diffraction data for a (weak)

high-order reflection (220) diffracting at 40.86� (2�). Fig. 7(b)

shows one slide of the diffraction data set revealing the (220)/

(022) double peak owing to pseudo-merohedral twinning. The

measured �2�(220) angle, 0.15�, is relatively close to the

expected one, 0.12�. Furthermore, the measured �� angle

between the two centres, 0.11�, is also fully consistent with the

twin description. Finally, it must be noted that curved fringes

research papers

J. Synchrotron Rad. (2010). 17, 751–760 Miguel A. G. Aranda et al. � Twinned microcrystals 757electronic reprint

connecting the two peak centres, see Fig. 7(b), were also

observed in other coherent X-ray diffraction data sets.

The consequences of twinning in the diffraction data sets

described above are also applicable to LPCMO microcrystals.

However, to distinguish between (121) and (200) type reflec-

tions is not straightforward because the very low Sac lattice

difference (see Table 1) makes the peak splitting very small for

both type of reflections. For LPCMO microcrystals, five data

sets were measured and two of them have clear split peaks.

Fig. 7(c) shows an image of a split peak. The measured

�2�(200) angle, 0.022�, is larger than the expected value, 0.010�

(see Table 1). The measured �� angle between the two

centres was 0.057�. Although the split angles are somewhat

larger that the expected ones, for the average composition,

these observations are totally consistent with this twin

description. Furthermore, this is to be expected because the

(average) composition of the studied microcrystal may be not

exactly the same as that of the powder.

Since the beam was made coherent during the experiment

by use of a coherence-defining aperture and then focused to

obtain sufficient intensity from the micrometre-sized crystals,

the speckles and fringes of the diffraction patterns of split

Bragg peaks were found to interfere with each other and give

a single merged coherent diffraction pattern. From the fringes

spacing, an estimation of the crystal sizes can be made,

Size ðnmÞ ’ �D=nps ’ 9455=n; ð4Þ

where D is the sample-to-detector distance, n is the average

number of the pixel for the repeating fringes, and ps is the

pixel size of the CCD detector. We have applied equation (4)

to many fringe sets such as those shown in Figs. 4 and 7. The

average sizes ranged between 450 and 900 nm for different

microcrystals. Sizes close to 700 nm were obtained very

frequently. These sizes are consistent with those observed by

electron microscopy (Fig. 2). It must also be noted that non-

regular fringe spacing has been observed for several crystals.

This could be due to very strained crystals as reported

previously (Cha et al., 2010).

It was observed that the split peaks, discussed above, have

different intensities. This can be explained by smaller or fewer

domains for one orientation. In order to study the possible

effects of the twin domain size, several simulations have been

carried out. Fig. 8 displays the three-dimensional coherent

diffraction patterns of three twinned objects modelled as

explained above. Fig. 8(a) displays the pattern for a twin pair

with a twin being half the size of its sister. Fig. 8(b) displays a

similar study but for a twin being four times smaller. Finally,

Fig. 8(c) shows the simulation for a twin being eight times

smaller than its sister. The phase ramp for these simulations is

the same as that employed in Fig. 6(a). The centre of symmetry

evident in Fig. 6 is lost, as expected, because the twins have

different sizes but the intermodulation signal between the two

peaks is retained. Furthermore, the diffraction peaks arising

from the twins with the smaller sizes are wider. As the domain

size decreases, with respect to its twin sister, its fringe spacing

becomes broader (see Fig. 8c) as, with only one domain of

each type, the fringe spacing is determined by the size of the

domains. This might be an alternative explanation for non-

regular fringe spacing.

In addition to the different size of the twin domains,

complex strain patterns may take place within the twinned

microcrystals. A TEM study of this work (see Fig. 2) and many

other reports about these types of manganites (see, for

instance, Uehara et al., 1999; Fath et al., 1999; Kim et al., 2000)

showed very complex microstructures and behaviours. The

coherent diffraction patterns showed a curved set of fringes

[see, for instance, Fig. 7(b)] and fringe branches without a

constant spacing. In order to justify these observations, we

have carried out simulation studies imposing a more complex

phase change within the particles. Fig. 9 shows three examples

of this type of simulation. In this case the phase is not constant

with respect to the faces of the microcrystal but they have a

curvature modelled by a parabolic function. The phase ramp is

similar to that of Fig. 6(a) and the magnitude of the curvature

has been varied. As can be seen, Fig. 9(a) has a small curvature

and its coherent diffraction pattern is quite similar to that

reported in Fig. 6(a). However, as curvature of the phase

research papers

758 Miguel A. G. Aranda et al. � Twinned microcrystals J. Synchrotron Rad. (2010). 17, 751–760

Figure 7Coherent diffraction images for (a) (200)/(002) twin pair (PCMO1–18, frame 21; 2 s of exposure time, five accumulations, Imax = 327670 counts s�1), (b)(220)/(022) twin pair (PCMO3–4, frame 33; 120 s of exposure time, one accumulation, Imax = 2330 counts s�1) and (c) likely (200)/(002) twin pair(LPCMO16, frame 43; 60 s of exposure time, one accumulation, Imax = 2440 counts s�1). Note the curved fringes that connect both twin peaks in (b). Thefull CCD recording area is shown for the sake of comparison.

electronic reprint

(strain) within the crystal increases (see

Fig. 9b) the intermodulation signal

increases in intensity. Furthermore,

when strain is more relevant (see Fig. 9c)

curvature of the fringes develops. The

interplay between non-flat faces, large

strain fields and different twin domain

sizes is believed to generate the

complex fringe/speckle patterns

observed in this study.

Finally, simulations were also carried

out in order to study the role of the

number of domains in the three-

dimensional coherent diffraction

patterns. Fig. 10 displays the coherent

diffraction signal corresponding to a

twinned crystal with four alternative

domains (ABAB). In addition to the

split peak, owing to the two types of

domains (A and B), three speckles are

evident within each diffraction maxima

(see bottom of Fig. 10) for this twin

arrangement. The simulations can be

interpreted as follows: (i) the number of

diffraction peaks is the number of

opposite phase ramps (two in this case);

(ii) the number of speckles within a

peak (three in this case) is the overall

number of domains minus one, the

speckles within each peak arising from

the ABA and BAB configurations,

respectively. This can be readily gener-

alized to n domains within a crystal but

the diffraction from a given specimen

may be complex owing to the possible

existence of domains related by

different twin laws, not only 180� rota-

tion around [101], and the complex

strain pattern within the microcrystal. A

multi-twinned microcrystal would give

complex three-dimensional patterns

such as that shown in Fig. 4(c).

In summary, using coherent X-ray

diffraction, Bragg peaks of pseudo-

merohedral twinned manganite micro-

crystals have been measured and they

were broken up into fringes and

speckles. This is consistent with a new

theory of the connection between the

twinning and the phase of the domain

that contributes to a given Bragg peak.

It is straightforward to show that the

number of speckles in the Bragg peak is

indicative of the number of domains in

the beam. Our new understanding of

the origin of this speckle shows that the

visibility is proportional to the phase

research papers

J. Synchrotron Rad. (2010). 17, 751–760 Miguel A. G. Aranda et al. � Twinned microcrystals 759

Figure 9(Top) Three bicrystals shown as a translucent three-dimensional box with colours representing thephase change. The two twins have the same dimension (24 � 24 � 24 grid points) and the sameoverall variation of the phase ramp. However, the curvature of the parabolic function changes in thethree objects. The change has been modelled by displacing the minimum of the parabolic functionby 78 (a), 35 (b) and 15 (c) pixels from the object. (Middle) Calculated three-dimensional coherentdiffraction patterns of the twinned objects. (Bottom) Selected isoscalar plane cut for the diffractionpatterns.

Figure 8(Top) Three bicrystals shown as a translucent three-dimensional box with colours representing thephase change which is the same as that given in Fig. 6(a). The two twins within each bicrystal havedifferent dimensions: (a) (24 � 24 � 24) and (24 � 12 � 24), (b) (24 � 24 � 24) and (24 � 12 � 12)and (c) (24 � 24 � 24) and (12 � 12 � 12) grid points. (Middle) Calculated three-dimensionalcoherent diffraction patterns of the two twinned objects. (Bottom) Isoscalar plane cut of thediffraction patterns.

electronic reprint

shift between the domains. This method represents an entirely

new way of measuring the domains mismatch and it may allow

sensitive testing of strain in crystals. In the future, successful

phase retrieval will allow mapping out the twin domains and

twin boundaries which play a key role in many physical

properties like magnetoresistance.

Diamond Light Source is thanked for the provision of X-ray

synchrotron diffraction beam time, and Gareth Nisbet for his

assistance during the experiment. We also thank James Holton

for providing a copy of the nearBragg program. The work at

Malaga was financed by MAT2009-07016 research grant and

at UCL under EPSRC grant EP/F020767/1 and an ERC

Advanced grant.

References

Alaimo, M. D., Potenza, M. A. C., Manfredda, M., Geloni, G., Sztucki,M., Narayanan, T. & Giglio, M. (2009). Phys. Rev. Lett. 103, 194805.

Born, M. & Wolf, E. (1999). Principles of Optics, 7th ed. CambridgeUniversity Press.

Buerger, M. J. (1960). Crystal Structure Analysis. New York: JohnWiley and Sons.

Cahn, R. W. (1954). Adv. Phys. 3, 363–445.

Cha, W., Song, S., Jeong, N. C., Harder, R., Yoon, K. B., Robinson,I. K. & Kim, H. (2010). New J. Phys. 12, 035022.

Collado, J. A., Frontera, C., Garcia-Munoz, J. L., Ritter, C., Brunelli,M. & Aranda, M. A. G. (2003). Chem. Mater. 15, 167–174.

Daoud-Aladine, A., Rodriguez-Carvajal, J., Pinsard-Gaudart, L.,Fernandez-Dıaz, M. T. & Revcolevschi, A. (2002). Phys. Rev. Lett.89, 097205.

Fath, M., Freisem, S., Menovsky, A. A., Tomioka, Y., Aarts, J. &Mydosh, J. A. (1999). Science, 285, 1540–1542.

Fienup, J. (1982). Appl. Opt. 21, 2758–2769.Hervieu, V., Van Tendeloo, G., Caignaert, V., Maignan, A. & Raveau,

B. (1996). Phys. Rev. B, 53, 14274–14284.Ji, G., Morniroli, J.-P., Auchterlonie, G. J., Drennan, J. & Jacob, D.

(2009). Ultramicroscopy, 109, 1282–1294.Jiang, H., Ramunno-Johnson, D., Song, C., Amirbekian, B., Kohmura,

Y., Nishino, Y., Takahashi, Y., Ishikawa, T. & Miao, J. (2008). Phys.Rev. Lett. 100, 038103.

Keller, L. P. & Buseck, P. R. (1994). Am. Miner. 79, 73–79.Kim, K. H., Uehara, M., Hess, C. & Sharma, P. A. (2000). Phys. Rev.Lett. 84, 2961–2964.

Koch, E. (1992). International Tables for Crystallography, Vol. C,edited by A. J. C. Wilson, pp. 10–14. Dordrecht: Kluwer.

Leake, S. J., Newton, M. C., Harder, R. & Robinson, I. K. (2009). Opt.Express, 17, 15853–15859.

Levy, P., Parisi, F., Granja, L., Indelicato, E. & Polla, G. (2002). Phys.Rev. Lett. 89, 137001.

Miao, J., Charalambous, P., Kirz, J. & Sayre, D. (1999). Nature(London), 400, 342–344.

Minkevich, A. A., Gailhanou, M., Micha, J.-S., Charlet, B., Chamard,V. & Thomas, O. (2007). Phys. Rev. B, 76, 104106.

Neutze, R., Wouts, R., Van der Spoel, D., Weckert, E. & Hajdu, J.(2000). Nature (London), 406, 752–757.

Newton, M. C., Leake, S. J., Harder, R. & Robinson, I. K. (2010). Nat.Mater. 9, 120–124.

Nishino, Y., Takahashi, Y., Imamoto, N., Ishikawa, T. & Maeshima, K.(2009). Phys. Rev. Lett. 102, 018101.

Pfeifer, M. A., Williams, G. J., Vartanyants, I. A., Harder, R. &Robinson, I. K. (2006). Nature (London), 442, 63–66.

Robinson, I., Gruebel, G. & Mochrie, S. (2010). New J. Phys. 12,035002.

Robinson, I. & Harder, R. (2009). Nat. Mater. 8, 291–298.Santoro, A. (1974). Acta Cryst. A30, 224–231.Sarma, D. D., Topwal, D., Manju, U., Krishnakumar, S. R., Bertolo,

M., La Rosa, S., Cautero, G., Koo, T. Y., Sharma, P. A., Cheong,S. W. & Fujimori, A. (2004). Phys. Rev. Lett. 93, 097202.

Sayre, D. (1952). Acta Cryst. 5, 843.Thibault, P., Dierolf, M., Menzel, A., Bunk, O., David, C. & Pfeiffer, F.

(2008). Science, 321, 379–382.Turner, J. J., Jordan-Sweet, J. L., Upton, M., Hill, J. P., Tokura, Y.,

Tomioka, Y. & Kevan, S. D. (2008). Appl. Phys. Lett. 92, 131907.Uehara, M., Mori, S., Chen, C. H. & Cheong, S. W. (1999). Nature(London), 399, 560–563.

Van Aken, B. B., Meetsma, A., Tomioka, Y., Tokura, Y. & Palstra,T. T. M. (2002). Phys. Rev. B, 66, 224414.

Vartanyants, I. A., Robinson, I. K., McNulty, I., David, C., Wochner, P.& Tschentscher, Th. (2007). J. Synchrotron Rad. 14, 453–470.

Wang, Y. & Liebermann, R. C. (1993). Phys. Chem. Miner. 20, 147–158.

White, T. J., Segall, R. L., Barry, J. C. & Hutchison, J. L. (1985). ActaCryst. B41, 93–98.

Yeates, T. O. (1997). Methods Enzymol. 276, 344–358.

research papers

760 Miguel A. G. Aranda et al. � Twinned microcrystals J. Synchrotron Rad. (2010). 17, 751–760

Figure 10(Top) Twinned crystal with four domains (ABAB) shown as a translucentthree-dimensional box with the domain dimensions and phase ramps asin Fig. 6(a). (Middle) Calculated three-dimensional coherent diffractionpattern. (Bottom) Isoscalar plane cut of the diffraction pattern showingthe speckles within the split peaks.

electronic reprint